Rough Sets Probabilistic Data Association Algorithm

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ScienceDirect Defence Technology 9 (2013) 208e216

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Rough Sets Probabilistic Data Association Algorithm and its Application in Multi-target Tracking Long-qiang NI a,*, She-sheng GAO a, Peng-cheng FENG a, Kai ZHAO b a

Department of Automatic Control, Northwestern Polytechnical University, Xi’an 710072, China b Northwest Institute of Mechanical & Electrical Engineering, Xianyang 721099, China Received 15 May 2013; revised 14 August 2013; accepted 19 August 2013 Available online 6 December 2013

Abstract A rough set probabilistic data association (RS-PDA) algorithm is proposed for reducing the complexity and time consumption of data association and enhancing the accuracy of tracking results in multi-target tracking application. In this new algorithm, the measurements lying in the intersection of two or more validation regions are allocated to the corresponding targets through rough set theory, and the multi-target tracking problem is transformed into a single target tracking after the classification of measurements lying in the intersection region. Several typical multi-target tracking applications are given. The simulation results show that the algorithm can not only reduce the complexity and time consumption but also enhance the accuracy and stability of the tracking results. Copyright Ó 2013, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved. Keywords: Rough set; Target tracking; Data association; Data fusion

1. Introduction In multi-target tracking (MTT) application, people want to estimate the states of several targets according to the measurements from some kinds of sensors (such as radar, infrared sensor and sonar). If the relationship between a tracked target and measurements lying in the validation regions is known beforehand, the target states could be estimated through standard filtering algorithms [1e5] (Kalman filter, extended Kalman filter and sample-based filtering methods). But in practical application, this situation may not exist: some of the measurements may be originated from clutter or false alarms (returns from nearby objects like building, water tower, mountain, rain or ECM). So it is * Corresponding author. E-mail address: [email protected] (L.Q. NI). Peer review under responsibility of China Ordnance Society

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the most important to determine the corresponding relationships between targets and measurements lying in the intersection of two or more validation regions in multi-target tracking [6]. Two methods can be used to solve the data association problem: a) unique-neighbor data association [7,8], such as global nearest neighbor standard filter (GNNSF) and multi-hypothesis tracking (MHT), which is used to associate all the measurements (or one of the measurements) with one of the previously created tracks; and b) all-neighbor data association [6,9,10], such as joint probabilistic data association (JPDA), which is used to update all tracks by using all the measurements lying in the validation region. GNNSF only considers the most likely hypothesis for track update and new track initiation so that it is only used for sparse targets, accurate measurements, and low false alarms in the validation region [8,11]. The multiple hypothesis tracker (MHT) is recognized theoretically as the optimum approach in Bayesian sense under idealized modeling assumptions. Unfortunately, the computational complexity of MHT limits its practical realization even using the fastest computers available [9,12]. In JPDA, a target tracking is updated by a weighted sum of all the measurements in the validation region [13e15]. This means that the

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L.Q. NI et al. / Defence Technology 9 (2013) 208e216

measurements may contribute to the update of more than one target tracks. And also JPDA increases the track variance matrix of Kalman filter to account for the association uncertainty, however the increased covariance matrix leads to even more false observations in the validation region [8]. In Ref. [9]，a new allneighbor fuzzy association approach is used for tracking multitarget in a clutter environment. This method has a lower computational complexity in a clutter environment at the expense of a lower performance compared to the standard JPDA. In this paper, the rough set theory is used to deal with the echo measurements in the validation regions, and the measurements in the lower approximation, including the measurements originated from target under discussion absolutely， are used to update the target states directly; the measurements, including the measurements which are not originated from target under discussion and may be originated from target under discussion, which lie in the upper approximation but are not in the boundary region (intersection region) are discarded for target under discussion; the measurements in the boundary region are approximated by the measurements in the lower approximation and upper approximation. In this approach，the measurements in the intersection region are handled discriminatingly, and the validation matrix needs not to be calculated. The approach can efficiently reduce the association uncertainty compared with the joint probability data association algorithm. It can reduce the computational complexity to a large degree, and also do not increase the track variance, especially in dense clutter environment.

2. Rough sets and the conformation of decision table for MTT We assume that there are two targets of interest in the filed of view (FOV), as shown in Fig. 1. In the two validation regions, the measurements y1 and, y2 may be originated from Target 1 or clutter; the measurementsy5, y6 and y7 may be originated from Target 2 or clutter; y3 and y4 may be originated form Target 1, Target 2 or clutter (clutters). If it can be decided that the measurements y3 and y4 belong to Target 1 or Target 2, the probabilistic data association (PDA) algorithm which can only be used for a single target tracking would be used to deal with the data association in MTT.

209

Unfortunately, the measurements lying in the intersection area y2 . Fuzzy uncertainty is alare of fuzzy uncertainty for b y1 and b ways associated with a boundary region of a set [16]. For example, if an element does not belong to a special set and also does not belong to its complementary set in domain of discourse, it probably belongs to the boundary region of the two sets. The rough set theory [17e22] is a novel mathematics method which can be used to process the uncertain, imprecise and fuzzy informations. This method is based on a classification mechanism, which is considered as equivalence (indiscernibility) relation in the given space. In MTT, the returns lied in the intersection of validation regions can be considered as the boundary elements, and these measurements are uncertain according with the association rule (relationship between targets and measurements). In the rough set theory, the knowledge is often expressed as information table (information system). The values of elements in the information table are often acquired from sensor or given by expert. In the rough set theory, an information system is denoted as a quadruple [22]: (S ¼ U, A, F, Vd), where U and A are the domain of discussion and the attribute set，respectively, which are the non-empty finite sets, F is the maping from set U to set A, and Vd is the value of attribute set. If the attribute set A is decomposed into C and D 3 A, C and D are called conditional attribute and decision attribute, respectively. The information systems are called the decision information systems. The process of data association in MTT is a typical decision information system in which the cumulative set of measurements at time index k can be regarded as the attribute set, and the association relation between target and measurement can be taken as decision attribute, which the measurement associates with which target. For example，in Fig. 1, for the domain of discourse U ¼ {y1,y2,y3,y4,y5,y6,y7}, the attribute set can be expressed as A ¼ fC; Dg ¼ ffV k ; di ðkÞ; RðkÞg; fassociationgg with respect to the decision attribute (association relation), we can get the following equivalence relation Rd ¼ ffy1 ; y2 g; fy3 ; y4 g; fy5 ; y6 ; y7 gg Since {y1,y2}, {y3,y4}, {y5,y6,y7} are indiscernible with respect to decision attribute, other information needs to be derived from this information system to refine the indiscernible relation. A decision table associated with Fig. 1 is given in Table 1. In Table 1, Table 1 Decision table for Fig. 1. Attribute Vk

Fig. 1. Two targets with common measurements.

y1 y2 y3 y4 y5 y6 y7

Eq. Eq. Eq. Eq. Eq. Eq. Eq.

(1) (1) (1) (1) (1) (1) (1)

Decision di(k)

R(k)

Eq. Eq. Eq. Eq. Eq. Eq. Eq.

Variance Variance Variance Variance Variance Variance Variance

(14) (14) (14) (14) (14) (14) (14)

Association matrix matrix matrix matrix matrix matrix matrix

t1 t1 t1/t2 t1/t2 t2 t2 t2

210


T Vk ¼ yi : ½yi b yðkÞ R1 ðkÞ½yi byðkÞ g T ¼ yi : ðei;k Þ R1 ðkÞei;k g

ð1Þ

where Vk is the validation region [23], and di2 ðkÞ (Eq. (14)) is the distance between the measurement i and the predicted

nz =2 (4) jSk j1=2 P Y k jqik ; mk ; Y k1 ¼ Vkmk þ1 P1 G ð2pÞ 1 ~ exp ~yTkjk1;i S1 y kjk1;i k 2

ð4Þ

P qik mk ; Y k1 ¼ P qik mk ¼ P qik M f ¼ mk 1; M t ¼ mk P M f ¼ mk 1M t ¼ mk h i1 8 mf ðmk Þ 1 > P P P þ ð1 P P Þ P D G D G D G < mf ðmk 1Þ mk þ P qik M f ¼ mk ; M t ¼ mk P M f ¼ mk M t ¼ mk ¼ h i1 > : m ðm Þ m ðm Þ ð1 PD PG Þ mf ðmf k k 1Þ PD PG þ ð1 PD PG Þ mf ðmf k k 1Þ ð5Þ

measurement, and R(k) is the covariance matrix of measurement error. 3. Rough sets probabilistic data association (RS-PDA) Probabilistic data association (PDA) is to use all of the validated measurements with different weights (probabilities) to update the target states [6,10,23]. The PDA algorithm is used to calculate the association probability which describes the contribution of each validated measurement to the target of interest in real time. The association probability used in a tracking filter accounts for the uncertainty of measurement origin. Since this method does not take into account the measurements in the intersection of two or more validation regions, it can be used only for the tracking of a single target in clutter environment. The association probabilitybik, which describes the probability of a measurement origin in PDA, is calculated as follows. Assume that qik is the association event which expresses whether the i-th measurement is associated with a target at time k, Yk is the set of validated measurements at time k, Yk is the cumulative set of measurements at time k, mk is the number of the validated measurements at time k, Vk is the volume of validation region, PG is the gate probability of that the correct measurement falls in the validation region, and PD is the detection probability of that the true measurement is detected. The association probability is calculated as follows [23] bik ¼ P qik Y k ¼ P qik Y k ; mk ; Y k1 ¼ 1c P Y k jqik ; mk ; Y k1 P qik mk ; Y k1

ð2Þ

where c¼

mk X P Y k jqik ; mk ; Y k1 P qik mk ; Y k1 i¼0

ð3Þ

It can be known from the calculation process that the association probability is related with the volume of validation region, number of validated measurements, gate probability, detection probability, distribution function of false measurements and measurement residuals. In multi-target tracking, the calculation of association probabilities is quite complicated because a measurement can be originated from more than one target (for example, the measurements lie in the intersection region). Therefore, the probabilistic data association filter (PDAF) is not suitable for multi-target tracking problem unless it can be determined which measurements in the intersection region are originated from which target. The rough set theory proposed by Pawlak, which is used to deal with the imprecise and uncertain information system with potential knowledge of the information system, is discussed. In the theory, the knowledge is processed and found using equivalence relation or granularity. The equivalence relation and granularity are expressed by two definable or observable subsets called lower and upper approximations. The rough set theory has been successfully applied to machine learning, intelligent system, inductive reasoning, pattern recognition, image processing, signal analysis, knowledge discovery, decision analysis, expert system and many other fields. In the rough set theory, no prior information beyond the information system to be discussed is needed to accomplish the inference compared with other theory. In multi-target tracking, the measurements lying in the intersection region are uncertain for association application, which makes PDAF unsuitable for MTT. The rough sets theory can be used to deal with the measurements inside the intersection region and approximate these measurements with the measurements certainly associated with some target to be discussed. Two operators are defined on set X 4 U, that is BðXÞ ¼ xi j½xB 4X

ð6Þ

BðXÞ ¼ xi j½xB X X s B

ð7Þ


For any given set X on universe U, BðXÞ and BðXÞ are called B-lower and B-upper approximations of set X respectively. Let BNB ðXÞ ¼ BðXÞ BðXÞ

ð8Þ

BNB(X) is called the boundary set of X. For example, two targets at time index k have an intersection validation region, and some measurements lie in the intersection of the two validation regions. According to Fig. 1, the lower approximation, upper approximation and boundary region of Target 1 are expressed in Fig. 2.

Fig. 2. Upper and lower approximations about measurements for validation regions.

In Fig. 2, for Target 1, the measurements in the lower approximation are equivalent, these measurements can be used to update the state of Target 1 directly, and the measurements in the upper approximation but not in the boundary region are discarded, and the measurements in the boundary region are indiscernible for Target 1 or Target 2. Therefore, for a given target, only part of the measurements can be used to update the target states. The key of data association is to find out the property of measurements in the boundary region in MTT application (the corresponding relationship between measurements and targets). The approximation quality of rough sets is expressed as follows [17,22]

card BðXÞ aB ðXÞ ¼ card BðXÞ

ð9Þ

obviously 0 aB(X) 1: 1) If aB(X) ¼ 1, then X is crisp with respect to B (X is precise with respect to B). That is to say, the boundary set BNB(X) ¼ 0, this indicate that no validation intersection about targets of interest exists in multi-target tracking; 2) If aB(X) < 1, then X is rough with respect to B (X is vague with respect to B). This indicates the intersection regions exist between the target of interest and other targets, and also there are some measurements in the intersection regions;

211

3) If aB(X) ¼ 0, then X is internally B-indefinable. This means that all the measurements of target of interest lie in the validation region of other targets. For 1), the single target tracking and data association approach can be used for multi-target tracking; for 2), the association relationship between measurements and targets needs to be found out, and the tracking filter and data association approach are used to update the target states, which will be emphatically discussed in this paper; for 3), the group and extended target tracking approaches can be used. Approximation quality (Eq. (9)) can be used to measure the quality of decision on universe U, so it is proper to use this parameter to approximate the measurements in the boundary region with the measurements in the lower approximation. In the data association, all the measurements in the validation region are approximated with the measurements in the lower approximation by approximation quality. Obviously, if the approximation quality is high, the measurements in the boundary region make more contribution to the target of interest; if the approximation quality is low, the measurements in the boundary region make less contribution to the target of interest. Therefore, the measurements in the boundary region can be allotted to the related targets according to this contribution factor. For the targets having intersection validation regions, the approximation quality factors are inversely arranged according to the target number. That is aiB ðXÞ ¼ aNi B ðXÞ

ð10Þ

The association probability of the measurements in the boundary region belonging to the associated targets can be expressed as PðNi Þ ¼ aiB ðXÞ =

N X

aiB ðXÞ

ð11Þ

i¼1

and the number of boundary region measurements belonging to the associated validation region of target t is Nit ¼ PðNi Þ cardðBNB ðXÞÞ

ð12Þ

After the number of targets associated with related targets is calculated from Eq. (12), the position of the measurements in the boundary region for the target of interest should be determined. To decide this distribution of measurements, here the measurements in the boundary region is considered as a whole, and the relative distance factor is calculated by Eq. (13). dt ðkÞ ¼

cardðBN XB ðXÞÞ i¼1

di2 ðkÞ 2 dmax;i ðkÞ

ð13Þ

di2 ðkÞ ¼ ei ðk þ 1Þ S1 ðk þ 1Þei ðk þ 1Þ

ð14Þ

b 1Þ ei ðkÞ ¼ Y i ðkÞ HXðkjk

ð15Þ

T

212


Where，di2 ðkÞ is the distance between the predicted measurement and the considered measurement; ei(k) is the mea2 surement residual; dmax;i ðkÞ is the maximum distance on the direction of line connecting between the predicted measurement and the considered measurement in the validation region. For Target t, Nit measurements are selected from the boundary region randomly, and the value of the relative distance factor dt(k) should be kept constant. After the number and position of the measurements in boundary region are determined, the updated states of target can be calculated as follows [6,10,23] b XðkjkÞ ¼

mðkÞ X

b i ðkjkÞ bi ðkÞX

wt ðkÞwNð0; QðkÞÞ

ð26Þ

The measurements are modeled by tan ½ðxðkÞ xs Þ=ðyðkÞ ysffiÞ qðkÞ YðkÞ ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þvðkÞ ðyðkÞ ys Þ þ ðyðkÞ ys Þ rðkÞ

ð27Þ

where v(k) is the measurement noise, which is assumed as white and normal probability distributions, that is vðkÞwNð0; RðkÞÞ

ð28Þ

ð16Þ

i¼0

4.2. Data simulation and analysis

where

mðkÞ ¼ Nkt þ card BðXÞ

ð17Þ

b i ðkjkÞ ¼ Xðkjk b X 1Þ þ KðkÞei ðkÞ ð18Þ b i ðkjkÞ is the updated state of the i-th validated measureX ment at time index k, K(k) is Kalman gain, here: K k ¼ P k k 1 HT k S1 k ð19Þ b P(kjk) is the estimated variance of XðkjkÞ, where ~ PðkjkÞ ¼ b0 ðkÞPðkjk 1Þ þ ½1 b0 ðkÞPc ðkjkÞ þ PðkÞ ð20Þ Pðk þ 1jkÞ ¼ FðkÞPðkjkÞFT ðkÞ þ QðkÞ

ð21Þ

2

3 1 Dt 0 0 6 0 1 Dt 0 7 7 Ft ðkÞ ¼ 6 4 0 0 1 Dt 5 0 0 0 1

1

Pc ðkjkÞ ¼ ½I KðkÞHðkÞPðkjk 1Þ "

mk X i¼1

ð22Þ #

bi ðkÞvi ðkÞvTi ðkÞ

vðkÞv ðkÞ K ðkÞ T

T

ð23Þ

4. Data simulation and result analysis

ð29Þ

For turning object, the coordinated turn model is used: 2

and

~ PðkÞ ¼ KðkÞ

In practical application, most of the target motion is composed of straight line and turning motions, including uniform speed or uniform acceleration, and the relative space relation of multi-target is composed of cross relation, parallel relation and split relation. For straight line moving target, the dynamics of the target is modeled as a linear discrete Wiener velocity model (DWVM):

sinðutk DtÞ

0

1cosðutk DtÞ

3

utk utk 6 7 6 7 6 0 cos ut Dt 0 sin ut Dt 7 6 7 k k 7 Ft ðkÞ ¼ 6 6 7 sinðutk DtÞ 6 0 1cosðutk DtÞ 1 7 t t 6 7 uk uk 4 5 t t 0 sin uk Dt 0 cos uk Dt

ð30Þ

and

4.1. Multi-target tracking model Target state is chosen as follows xðkÞ ¼ ½x x_ y y_

T

ð24Þ

where x; y are the positions of target are, and x_ ; y_ are the velocities of target. The dynamic model of targets is described as xt ðk þ 1Þ ¼ Ft ðkÞxt ðkÞ þ Gt ðkÞwt ðkÞ t ¼ 1; 2

ð25Þ

where Ft(k) is the state transition matrix, Gt(k) is the noise matrix, and wt(k) is the process noise which is assumed as white and normal probability distributions, that is

Fig. 3. Tracking results of two cross targets simulated with RS-PDAF (detailed view).


213

Fig. 4. Track errors of two cross targets simulated with RS-PDAF (50 Monte Carlo simulations, and total time consuming: 28.326 s).

Fig. 5. Tracking errors of two cross targets simulated with JPDAF (50 Monte Carlo simulations, and total time consuming: 35.536 s).

3 0 Dt2 =2 6 Dt 0 7 7 Gt ðkÞ ¼ 6 4 0 Dt2 =2 5 0 Dt 2

ð31Þ

Here RS-PDAF is used to track two targets. Fig. 3 shows the tracking results of two cross targets, Fig. 6 shows the tracking results of two parallel targets, and Fig. 9 shows the tracking results of two split targets in clutter environment. In Figs. 4,5,7,8,10 and 11, the tracking errors for two cross targets, parallel targets and split targets simulated by RS-PDAF and JPDAF are compared, respectively. Here we assume the sampling interval Dt ¼ 1 s, the simulation process lasts for 500 s, the clutter density (expected number of false returns per unit volume in the measurement space) l ¼ 103/km2, the measurement noise covariance matrix R ¼ [9 100], the process noise covariance matrix Q ¼ [16 16].

Fig. 6. Tracking results of two parallel targets simulated with RS-PDAF (detailed view).

214


Fig. 7. Tracking error of two parallel targets simulated with RS-PDAF (50 Monte Carlo simulations, and total time consuming: 27.764 s).

Fig. 8. Tracking errors of two parallel targets simulated with JPDAF (50 Monte Carlo simulations, and total time consuming: 34.562 s).

Fig. 9. Tracking results of two split targets simulated with RS-PDAF (detailed view).

Figs. 3e11 show the simulation results of cross, parallel and split targets based on RS-PDA and JPDAF, respectively. The tracking errors are listed in Table 2, and the time consuming is listed in Table 3. It can be seen from Tables 2 and 3 that the calculating errors of RS-PDA are less than those of JPDA, and the time consuming of RS-PDAF is 20% less than that of JPDAF. This is because JPDA makes an indiscriminate advantage of the measurements lying in the validation region. Therefore the measurements lying in the intersection region may be used for the state update of multiple targets, but RS-PDA distinguishes the measurements lying in the intersection of the validation regions for corresponding targets, compared with JPDAF this reduces the uncertainty about the origination of measurements. Because the RS-PDAF does not have to calculate the validation matrix which is time-consuming while JPDA filter does, the time consuming of RS-PDAF is less than that of JPDA filter.


215

Fig. 10. Tracking errors of two split targets simulated with RS-PDAF (50 Monte Carlo simulations, and total time consuming: 28.403 s).

Fig. 11. Tracking errors of two split targets simulated with JPDAF (50 Monte Carlo simulations, and total time consuming: 32.922 s).

5. Conclusions In this paper, a new data association method, RS-PDAF, based on rough sets theory is proposed. The method can reduce the uncertainty of the origination of measurements, and also make advantage of the measurements lying in the intersection region more objectively than JPDAF. After the

classification of rough sets, the relation between targets and measurements become clear so that the multi-target problem is easily translated into single target tracking. Compared with JPDAF, RS-PDAF reduces the number of measurements lying in the intersection region for a given target, makes data association process simpler and time-saving, and does not add the variance matrix of Kalman filter. Some typical data

Table 2 Tracking error comparison between two methods for three cases. Limit of error

Case

Method

Two cross targets

Two parallel targets

Two split targets

Two cross targets

Two parallel targets

Two split targets

Target 1

Target 2

Target 1

Target 2

Target 1

Target 2

x: y: x: y:

x: y: x: y:

x: y: x: y:

x: y: x: y:

x: y: x: y:

x: y: x: y:

RS-PDA JPDA

[12,3] [14,5] [14,7] [19,5]

[14,5] [15,7] [14,5] [17,5]

[13,4] [15,7] [20,7] [19,5]

[14,7] [12,4] [22,4] [20,8]

[14,6] [16,4] [18,9] [17,4]

[13,2] [14,3] [16,8] [15,4]

216


Table 3 Time consuming for 3 cases. Time consuming

Cross targets/s

Parallel targets/s

Split targets/s

RS-PDA JPDA

28.326 35.536

27.764 34.562

28.403 32.922

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